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Theorem cleljust 2115
Description: When the class variables in definition df-clel 2774 are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 2108 with the class variables in wcel 2107. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 2052 in order to remove dependencies on ax-10 2135, ax-12 2163, ax-13 2334. Note that there is no disjoint variable condition on 𝑥, 𝑦, that is, on the variables of the left-hand side, as should be the case for definitions. (Revised by BJ, 29-Dec-2020.)
Ref Expression
cleljust (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem cleljust
StepHypRef Expression
1 elequ1 2114 . . 3 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
21equsexvw 2052 . 2 (∃𝑧(𝑧 = 𝑥𝑧𝑦) ↔ 𝑥𝑦)
32bicomi 216 1 (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386  wex 1823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824
This theorem is referenced by:  bj-dfclel  33460
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